Hofstadter butterflies in macroscopic time-reversal invariant systems: Tight-binding electrons under n 2 × n 2 staggered magnetic fields

摘要: Hofstadter’s diagrams the energy spectra plotted against magnetic flux strength of tight-binding lattice electrons under n 2 staggered fields are obtained for various n’s. From n=1–14, these butterflylike continuous spectrum exhibit a systematic evolution approaching fractal Hofstadter butterfly spectrum. For larger n’s, butterflies can be roughly divided into two kinds regions, “Hofstadter-type regions” which bear structures, and “non-Hofstadter-type composed almost equally spaced subbands. The properties electronic states in non-Hofstadter-type regions revealed detail.